3.775 \(\int \frac{1+\sqrt{x}}{\left (1+\sqrt [3]{x}\right ) \sqrt{x}} \, dx\)

Optimal. Leaf size=42 \[ \frac{3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

[Out]

6*x^(1/6) - 3*x^(1/3) + (3*x^(2/3))/2 - 6*ArcTan[x^(1/6)] + 3*Log[1 + x^(1/3)]

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Rubi [A]  time = 0.241455, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + Sqrt[x])/((1 + x^(1/3))*Sqrt[x]),x]

[Out]

6*x^(1/6) - 3*x^(1/3) + (3*x^(2/3))/2 - 6*ArcTan[x^(1/6)] + 3*Log[1 + x^(1/3)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 \int ^{\sqrt{x}} \frac{\sqrt{x^{2}} + 1}{\sqrt [3]{x^{2}} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x**(1/2))/(1+x**(1/3))/x**(1/2),x)

[Out]

2*Integral((sqrt(x**2) + 1)/((x**2)**(1/3) + 1), (x, sqrt(x)))

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Mathematica [A]  time = 0.0183225, size = 42, normalized size = 1. \[ \frac{3 x^{2/3}}{2}-3 \sqrt [3]{x}+6 \sqrt [6]{x}+3 \log \left (\sqrt [3]{x}+1\right )-6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Sqrt[x])/((1 + x^(1/3))*Sqrt[x]),x]

[Out]

6*x^(1/6) - 3*x^(1/3) + (3*x^(2/3))/2 - 6*ArcTan[x^(1/6)] + 3*Log[1 + x^(1/3)]

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Maple [A]  time = 0.006, size = 48, normalized size = 1.1 \[ \ln \left ( 1+x \right ) +{\frac{3}{2}{x}^{{\frac{2}{3}}}}+2\,\ln \left ( 1+\sqrt [3]{x} \right ) -\ln \left ({x}^{{\frac{2}{3}}}-\sqrt [3]{x}+1 \right ) -3\,\sqrt [3]{x}+6\,\sqrt [6]{x}-6\,\arctan \left ( \sqrt [6]{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x^(1/2))/(1+x^(1/3))/x^(1/2),x)

[Out]

ln(1+x)+3/2*x^(2/3)+2*ln(1+x^(1/3))-ln(x^(2/3)-x^(1/3)+1)-3*x^(1/3)+6*x^(1/6)-6*
arctan(x^(1/6))

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Maxima [A]  time = 0.758112, size = 41, normalized size = 0.98 \[ \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \arctan \left (x^{\frac{1}{6}}\right ) + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x) + 1)/(sqrt(x)*(x^(1/3) + 1)),x, algorithm="maxima")

[Out]

3/2*x^(2/3) - 3*x^(1/3) + 6*x^(1/6) - 6*arctan(x^(1/6)) + 3*log(x^(1/3) + 1)

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Fricas [A]  time = 0.280949, size = 41, normalized size = 0.98 \[ \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \arctan \left (x^{\frac{1}{6}}\right ) + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x) + 1)/(sqrt(x)*(x^(1/3) + 1)),x, algorithm="fricas")

[Out]

3/2*x^(2/3) - 3*x^(1/3) + 6*x^(1/6) - 6*arctan(x^(1/6)) + 3*log(x^(1/3) + 1)

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Sympy [A]  time = 18.2063, size = 39, normalized size = 0.93 \[ 6 \sqrt [6]{x} + \frac{3 x^{\frac{2}{3}}}{2} - 3 \sqrt [3]{x} + 3 \log{\left (\sqrt [3]{x} + 1 \right )} - 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x**(1/2))/(1+x**(1/3))/x**(1/2),x)

[Out]

6*x**(1/6) + 3*x**(2/3)/2 - 3*x**(1/3) + 3*log(x**(1/3) + 1) - 6*atan(x**(1/6))

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GIAC/XCAS [A]  time = 0.261293, size = 41, normalized size = 0.98 \[ \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 6 \, x^{\frac{1}{6}} - 6 \, \arctan \left (x^{\frac{1}{6}}\right ) + 3 \,{\rm ln}\left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x) + 1)/(sqrt(x)*(x^(1/3) + 1)),x, algorithm="giac")

[Out]

3/2*x^(2/3) - 3*x^(1/3) + 6*x^(1/6) - 6*arctan(x^(1/6)) + 3*ln(x^(1/3) + 1)